An explicit finite volume method for viscoelastic fluid flows

We discuss a finite volume method for computing solutions of steady incompressible viscoelastic fluid flows. A fourth-order Runge-Kutta method is used in the explicit time-stepping scheme. The computations are carried out mainly on unstructured grids on Newtonian, inelastic and differential-type constitutive equations, which include the Oldroyd-B and the upper-convected Maxwell models. The performance of the scheme on unstructured grids is investigated, with particular reference to the stick-slip problem for the modified upper-convected Maxwell fluid. The results are compared with those obtained by using the finite element method whenever possible.This research was supported by the Australian Research Council (ARC).     

On heat transfer to pulsatile flow of a viscoelastic fluid

Summary A study is made of a problem of heat transfer to pulsatile flow of a viscoelastic fluid between two parallel plates of which the upper one is at a temperature higher than the lower one. The solutions for the steady and the fluctuating temperature distributions are obtained. The rate of heat transfer at the plates is also calculated. Numerical solutions are discussed with graphical representations. It is shown that the elasticity of the fluid significantly increases the temperature in the boundary layers near the plates. The magnitude of heat transfer at the plates is also greatly affected by the elasticity of the fluid and the Eckert number.     

A finite element/nodal volume technique for flow simulation of injection pultrusion

In this paper, a finite element/nodal volume technique is developed to simulate the resin flow through the reinforcement during injection pultrusion processes. The governing equations for the pressure distribution and the conservation of resin mass during injection pultrusion are first derived. The solution algorithm and its numerical implementation are described. In particular, an algorithm is developed to advance the flow front by taking into account both the resin flow relative to the reinforcement and the movement of the pultruded part as a whole. The numerical technique developed is validated against a one-dimensional analytical solution derived. A number of numerical tests are then conducted to investigate the numerical performance and capability of the technique.     

Unsteady oscillatory stagnation-point flow of a viscoelastic fluid

The unsteady stagnation point flow of the Walters B^′ fluid is examined and solutions are obtained. It is assumed that the infinite plate at y=0 is oscillating and the fluid impinges obliquely on the plate.     

Slow viscoelastic fluid flow past a rotating sphere

The problem of axially symmetric flow of a particular type of non-Newtonian fluid past a rotating sphere due to a uniform stream at infinity is investigated. The presence of a region of reversed flow is found under certain conditions depending on the angular velocity of the sphere, the speed of the uniform stream and radius of the sphere. This region which is attached to the rear portion of the sphere is found to depend strongly on the viscoelasticity of the fluid. The vortex is seen to move towards the sphere as the viscoelastic parameter increases while the other parameters are kept fixed. As this viscoelastic parameter approaches a critical value, the vortex is found to disappear.     

A finite volume procedure for fluid flow,heat transfer and solid-body stress analysis

A unified cell-centered unstructured mesh finite volume procedure is presented for fluid flow, heat transfer and solid-body stress analysis. An in-house procedure (A. W. Date, Solution of Transport Equations on Unstructured Meshes with Cell-Centered Colocated Variables. Part I: Discretization, International Journal of Heat and Mass Transfer, vol. 48 (6), 1117-1127, 2005) is extended to include the solid-body stress analysis. The transport terms for a cell-face are evaluated in a structured grid-like manner. The Cartesian gradients at the center of each cell-face are evaluated using the coordinate transformation relations. The accuracy of the procedure is demonstrated by solving several benchmark problems involving different boundary conditions, source terms, and types of loading.     

MHD flow of a viscoelastic fluid past a stretching surface

Summary The flow of a viscoelastic fluid past a stretching sheet in the presence of a transverse magnetic field is considered. An exact analytical solution of the governing non-linear boundary layer equation is obtained, showing that an external magnetic field has the same effect on the flow as the viscoelasticity.     

The flow of a viscoelastic fluid past a porous plate

Summary The flow of a second order viscoelastic fluid past a porous plate is considered. It is characterized by a boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. The boundary value problem is solved by making a plausible assumption, namely that the variation of the normal derivative of the velocity at the plate withk is sufficiently smooth, wherek is the viscoelastic fluid parameter. Under this assumption it is shown that dual solutions exist for values ofk less than a critical value. Beyond this value, no solution exists.     

Film flow of a nonlinear viscoelastic fluid along a conical rotor

Film flow of a nonlinear viscoelastic fluid, whose deformed behavior is described by using kinematic matrices, is considered along the surface of a rotating conical rotor.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 31, No. 2, pp. 231–236, August, 1976.     

Dispersion of a solute in a Poiseuille flow of a viscoelastic fluid

Gill and Sankarasubramanian's analysis of the dispersion of Newtonian fluids in laminar flow between two parallel walls is extended to the flow of non-Newtonian viscoelastic fluids (known as Phan-Thein-Tanner (PTT)). Using a generalized dispersion model which is valid for all times after the solute injection, the diffusion coefficient K_i(t) is obtained exactly and numerically for linearized and exponential forms of the PTT fluids, respectively. The analysis leads to the novel result for K₁ and K₂(t) (which is a measure of the longitudinal dispersion coefficient of the solute). It is found that the value of K₂(t) depends on the value of Deborah number (De=a measure of the level of elasticity in the fluid) whereas the value of K₁ is constant in both cases. Finally, the effect of the Deborah number on the axial distribution of the mean concentration θ_m is investigated in detail.     

Resistance of a sphere in a slow flow of a viscoelastic fluid

The authors have obtained an approximate solution of the problem of the resistance of a rigid sphere in a slow flow of a Maxwell viscoelastic fluid that is in good agreement with experimental data [1] for Weissenberg numbers We ≤ 0.7. It is shown that the effect of a decrease in the coefficient of resistance of a sphere in the interval 0.1 ≤ We ≤ 0.7 established experimentally is determined in full measure by the linear viscoelastic properties of the Maxwell fluid. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 6, pp. 1138–1140, November–December, 1998.     

Heat transfer in the flow of a viscoelastic fluid over a stretching sheet

Summary The problem of heat transfer in the viscoelastic fluid flow over a stretching sheet is examined. The important physical quantities such as the skin-friction coefficient and the heat transfer coefficient, are determined. It is found that the heat transfer coefficient decreases with the non-Newtonian parameter.     

MHD flow of a viscoelastic fluid past a stretching sheet with suction

Summary The laminar flow of a viscoelastic fluid past a stretching sheet in the presence of a magnetic field, when the fluid is extracted from the sheet at a uniform rate, is considered. An exact analytical solution exists for the problem. It is shown that when there is a suction of the fluid, the solutions are possible only upto a critical value of the viscoelastic parameter. Also, for values less than this critical value, dual solutions exist.     

The nonuniqueness of the MHD flow of a viscoelastic fluid past a stretching sheet

Summary Two closed-form solutions were found for the boundary layer equations of the title problem. Discussions are made to trace among them the physically realistic solution.     

On the stability of MHD flow of a viscoelastic fluid past a stretching sheet

Summary The effects of a magnetic field on the stability characteristics of a viscoelastic fluid flow due to the stretching sheet are investigated. A three-dimensional linear stability analysis is performed by means of the Method of Weighted Residuals for disturbances of the Taylor-Görtler type. It is found that the magnetic field exerts a stabilizing influence on the flow.     

Stagnation point flow and heat transfer for a viscoelastic fluid impinging on a quiescent fluid

A theoretical study is made in the region near the stagnation point when a lighter incompressible viscoelastic fluids impinges orthogonally on the surface of another quiescent heavier incompressible viscous fluid. Similarity solutions of the momentum balance equations for both fluids are equalized at the interface. It is noted that an exact boundary layer solution is obtained for the lower lighter fluid. The velocity of the lower fluid is independent of lateral interface velocity but the velocity of the upper viscoelastic fluid increases with increasing lateral interface velocity. It is observed that lateral interface velocity increases with increasing viscoelastic parameter for fixed values of density and viscosity ratio of the two fluids. The convective heat transfer is investigated base on the similarity solutions for the temperature distribution of the two fluids. The interface temperature increases with increasing viscoelastic parameter of the upper viscoelastic fluid.     

A nonintrusive flow measurement technique to validate the simulated laminar fluid flow in a packed container with vented walls

A nonintrusive flow measurement technique (particle image velocimetry) was used to determine the airflow field in a package with a container-to-product diameter ratio of less than 10. The complexity and uneven distribution of the measured flow field supported the requirement of a geometrical and mathematical model capable of describing the geometry and physics of flow within the package. Using novel computational fluid dynamics (CFD) codes, an accurate model of the packed structure was developed and the 3D Navier–Stokes equations were solved. A good agreement was obtained between experimental and predicted velocities. The detailed insight on the airflow pattern provided by the CFD analysis makes this approach an ideal tool to analyze the effect of different vent designs in the airflow field distribution in complex packaging systems.     

A finite boundary method for fluid flow field computations

A powerful new finite boundary concept of seeking field solution, at few selected regions in the solution domain, is introduced. Also a highly economical finite boundary method (FBM) which would greatly reduce the size of the coefficient matrix of the resulting system of simultaneous algebraic equations, requiring lesser computer memory and lesser computing time, is developed. Fluid flow fields governed by the basic elliptic partial differential equations—the Laplace's and the Poisson's equations—in two independent variables are mainly considered. The computational merits of the FBM are shown by solving, as an example, a simple representative flow problem, and the relevant computational finite boundary formulae are given in tabular form. The formulae are numerically derived based on a generalized method presented here. The added feature of the FBM is that it proves to be equally economical even when the solution is sought in the entire flow domain. The problem of steady-state viscous flows governed by the Navier-Stokes equations, the system of two simultaneous partial differential equations—the Poisson's equation and the vorticity transport equation—makes the FBM doubly economical. The possibility of developing an efficient hybrid computational algorithm, for curved problem boundaries, in conjunction with the finite element method, is discussed. The extension of FBM to transient, non-elliptic problems and to three-dimensional problem fields is also indicated. The FBM has been discussed in a more detailed manner so as to clearly bring out the advantages of the new finite boundary concept.     

State space formulation to viscoelastic fluid flow of magnetohydrodynamic free convection through a porous medium

Summary In this work we formulate the state space approach for one-dimensional problems of viscoelastic magnetohydrodynamic unsteady free convection flow through a porous medium past an infinite vertical plate. Laplace transform techniques are used. The resulting formulation is applied to a thermal shock problem and to a problem for the flow between two parallel fixed plates both without heat sources. Also a problem with a distribution of heat sources is considered. A numerical method is employed for the inversion of the Laplace transforms. Numerical results are given and illustrated graphically for the problem considered.Notation C specific heat at constant pressure - g acceleration due to gravity - density - time - u velocity component parallel to the plate - H _x induced magnetic field - x, y coordinates system - T temperature distribution - T _o temperature of the plate - T temperature of the fluid away from the plate - ₀ limiting viscosity at small rates to shear - v _o ^* / - v _m magnetic diffusivity - Alfven velocity - ^* coefficient of volume expansion - thermal conductivity - ^* thermal diffusivity - G Grashof number - Pr Prandtl number - L some characteristic length - k _o the elastic constant - K permeability of the porous medium